We consider the same setting as [1]}.
We consider a target measure \(\in (^d)\)  and the potential \(\mapsto (,)\) , as one would do in e.g. generative learning in imaging [2]}.
The measure \(\)  represents the model we train, parameterized as \(_\theta = \sum _i^n r_i^2 _{x_i}\)  with parameter \(\theta = ((x_i, r_i))_i\in (^2\times _+)^n\) .
Minimizing \((\cdot ,)\)  amounts to run a “standard” gradient update w.r.t. \(x_i\) , and a mirror gradient update w.r.t. \(r_i\) ., which reads
\(x_i^{(t+1)} & = x_i^{(t)} - \eta _x \nabla _{x_i} (_\theta ^{(t)}, ), \\r_i^{(t+1)} & = r_i^{(t)}.\exp \big ( - 2\eta _r \nabla _{r_i} (_\theta ^{(t)}, ) \big ),\) 
